Emergent Mathematics pt.3

f(x) roles = pigeonholes

While playing and studying Pokemon Black/White's battle system and developing metagame, I had a curious thought: Why are there tier levels? Is it possible for a game like Pokemon to have only one large tier? Surely, we already know the answers to these questions. There could be one large tier, but it wouldn't feature much variety. The uber Pokemon are significantly stronger than the lower tier Pokemon (see tier list here). For anyone who wants to win, it would be a severe disadvantage not to use ubers. So, most battlers would pick the uber Pokemon. This means out of 649 Pokemon, only about 20 or so would be viable. So, by making several different tiers (ubers, over used, under used, rarely used, not fully evolved, and little cup) players can choose a tier level and play with Pokemon that would never be viable in one large tier. The result is each tier has its own unique metagame and strategies. Battlers can find a place to compete with their favorite Pokemon. And I wouldn't have it any other way.

Sorry Pidove. There just aren't enough normal-flying slots for you. 

But this doesn't explain why things have to be this way even for players who favor diversity. Why can't all the Pokemon compete against each other? The simple answer is because the developers made some Pokemon really strong and others really weak. But really, this is just another way of saying that there's a clearly defined, functional reason behind the tiers. So, completely ignoring any kind of "developer defense," we have to look at the function of Pokemon battling for our answer. The reason some Pokemon can't compete is because competition is a game, games have goals/rules, and these parameters create the value scale with which we measure all battling Pokemon. For Pokemon, a great candidate has more than just high stat numbers overall. Typing, ability, move pool, and the metagame are important considerations. Remember, the Pokemon design space is filled with wrinkles that make evaluating Pokemon viability more interesting.  

You would think with all of these factors, that the battling value scale of Pokemon would be too complex to grasp. But it's really pretty simple. To understand, consider the design space of Pokemon. Yes, there are a lot of factors to consider as to what makes a good Pokemon and there are a huge range of moves, items, etc., but in the end players can only have up to 6 Pokemon on a team with 4 moves each. These two rules are the most significant limiting factors that allow the tiers to emerge. All you have to do is consider how a player would maximize their advantages.

With only 6 types of Pokemon you can only have a maximum of 12/17 types represented on your team. This means you already have offensive and defensive limitations and weak points. To best compensate for this weaknesses, you'll probably pick the Pokemon that have the best type coverage for attacking (i.e. few Pokemon resist their hits). Even if you load up your team with great attackers, to get the most bang for your buck, you have to maximize the attack strength of your Pokemon through special training. If you do this, you'll likely pick whichever attack stat (physical or special) is higher for each of you Pokemon. 

However, maximizing your offense creates even bigger holes in your defensive potential. Furthermore, some of the best attacking type coverage are the types of fire, water, and grass which are strong against each other in a Rock Paper Scissor like interplay loop and strong against many other types. After a while, you'll realize that rolling the dice and hoping that the opponent doesn't have the attacking types that counter your team isn't good enough. So to increase your chances of winning, you'll begin to switch out some of your attackers with Pokemon that have great defensive abilities but poor offense. You'll maximize their defense and rely on non damaging attacks or other special strategies to give your team more strategic versatility.

I could go on and on describing the step by step process of how Pokemon metagame develops. But the bottom line is that some of the most effective strategies (at least initially) are the ones that are less risky, more well-rounded in terms of offense and defense, and more versatile strategy-wise. So, because most battlers will look for Pokemon that are great at a few roles like leads, offense, walls/defense, sweepers, annoyers, etc. (see video here about Pokemon roles), you don't have 34 or 17 different slots to consider. You have much fewer! You want the best Pokemon that can fulfill these roles. These role "slots" determine how many Pokemon can fit within a tier without overlapping in function. For those who do overlap and are functionally inferior according to the value scale, they will be pushed to a lower tier. This is the basic idea of tier capacity.

The following is an except from wikipedia on the mathematical principle that inspired my theory.

In mathematics and computer science, the pigeonhole principle states that if n items are put into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. This theorem is exemplified in real-life by truisms like "there must be at least two left gloves or two right gloves in a group of three gloves". It is an example of a counting argument, and despite seeming intuitive it can be used to demonstrate possibly unexpected results; for example, that two people in London have the same number of hairs on their heads (see below).

To translate, Pokemon are the items and the number of pigeonholes are created from the game rules. So, part of the theory explains why it is impossible to have one large tier if the number of Pokemon available exceed the number of functional role slots. Even if you designed the differences between Pokemon to be nearly negligible, as long as there is a difference and n>m, there will be tiers. Some Pokemon will ultimately have higher stats and therefore fulfill a very specific role better than others. Using the pigeonhole principle, we can understand how a game emerges from its design space elements to the actual metagame. 

The ideas of tier capacity and design space holes/pigeonholes can be applied to single player games as well. If a game has many items like the guns in Metal Gear Solid 4 or in Borderlands, you can get a better idea of just how popular many of the items will be in actual use. This is partially an issue of balance, but more so an issue of metagame. After all, a game can be balance for a high level of play, but until the competitors get to that level various elements may be overused or seem unbalanced. 

I named this series emergent mathematics because there are numbers involved, some math-like calculations, and actual mathematical arguments used. I'm still researching how the ideas can be applied to game designers and researchers alike. In the future, I hope to provide clearer examples of these concepts at work. Until then... math on!